This thesis makes three theoretical contributions to the robust system analysis and control theory.
First, we revisit a central theorem in robust control theory known as the the Kalman-Yakubovich-Popov (KYP) lemma, and uncover its "hidden" symmetric structure that is rarely articulated in the literature of robustness analysis. Roughly speaking, we propose a new formulation of the KYP lemma so that two seemingly different quantities, "frequency" and "system uncertainties" play symmetric roles in the robust stability analysis. It turns out that the new formulation has sufficient generality to unify some of the recent extensions of the KYP lemma. Further consideration of this symmetry naturally leads us to the notion of mutual losslessness, which is the exact condition for the lossless of the analysis. As a result, the new formulation provides a general framework that answers when the KYP-like robustness analysis is lossless.
Second, we restrict our focus to the class of cone-preserving linear dynamical systems. Square MIMO transfer functions in this class have what we call the DC-dominance property: the spectral radius of the transfer function attains its maximum at zero frequency and hence, the stability of the interconnected transfer functions is guaranteed solely by the static gain analysis. Using this property, we prove the delay-independent stability of cone-preserving delay differential equations. This provides an alternative proof of the delay-independent mean-square stability of multi-dimensional geometric Brownian motions. Finally, we further restrict our focus to the special class of cone-preserving systems known as positive systems. We prove a novel "diagonal" KYP lemma for positive systems, which ensures the existence of a diagonal storage function without introducing conservatism whenever the system is contractive. This result suggests that a certain class of distributed optimal control for positive systems can be found via the semidefinite programming (SDP).